The Valuation of Corporate Debt with Default Risk

Hassan Naqvi∗

Financial Markets GroupLondon School of Economics

9th October 2002

Abstract. This article values equity and corporate debt by taking into accountthe fact that in practice the default point differs from the liquidation point and thatit might be in the creditors’ interest to delay liquidation. The article develops acontinuous time asset pricing model of debt restructuring which explicitly considersthe inalienability of human capital. The study finds that even though in generalthe creditors will not liquidate the firm on the incidence of default, but neverthelesswould liquidate the firm prematurely relative to the first best threshold. This agencyproblem leads to the breakdown of the capital structure irrelevance result.

1. Theoretical FoundationThe literature on the pricing of defaultable bonds started with Merton (1974) who appliedthe contingent claims valuation insight of Black and Scholes (1973) to the pricing ofcorporate bonds. He obtained closed form valuation expressions for (zero-coupon) riskybonds, by taking the lower reorganization boundary as given. Thus in his model, defaultoccurred at maturity if the value of the firm was less than the payment promised to thebondholders. Furthermore, at maturity the compensation received by the creditors is fixedand they receive the minimum of the value of the firm or the contracted payment. In theevent of a default, the control of the firm is transferred to the creditors and this defaultpoint can also be interpreted as a liquidation point with zero bankruptcy costs. Thus, inMerton’s model, both the lower reorganization boundary and the compensation receivedby the bondholders is taken to be exogenous. In practice, however, the compensationreceived by the creditors may vary continuously with the value of the firm if the firm isin financial distress. Furthermore, Merton does not make the distinction between defaultand liquidation.There have been a number of extensions of the Merton model. Black and Cox (1976)

incorporate bond indenture provisions. Jones, Mason and Rosenfeld (1984) consider mul-tiple issue of callable coupon debt. In both of these models however the induced lowerboundary at which the firm is liquidated is taken to be exogenous. Further, default isagain tantamount to bankruptcy in these models.Leland (1993) and Leland and Toft (1996) extend Merton’s analysis by endogenising

the lower reorganization boundary. By assuming that the equityholders will always issueequity to prevent a default, they obtain expressions for the values of risky debt via thesmooth-pasting condition. Thus in their models, the equityholders keep on issuing equity(when necessary) to avoid a default until the value of equity falls to zero. Thus defaultoccurs when the value of equity falls to zero and this default point is again synonymous

∗I would like to thank Ronald Anderson for his support and encouragement. I would also like tothank Hyun Song Shin and Abhijeet Singh for valuable comments. Comments from the participants ofFMG doctoral seminar are also appreciated. Any errors remain my responsibility. Correspondence tobe addressed to Hassan Naqvi, Room R4z17A, Financial Markets Group, London School of Economics,Houghton Street, London, WC2A 2AE, or email: s.naqvi@lse.ac.uk

1

The Valuation of Corporate Debt with Default Risk 2

with bankruptcy. They do admit that default need not lead to bankruptcy in practice,but nevertheless they interpret their default point as the bankruptcy point. Since theirdefault and bankruptcy points are in effect the same, the compensation received by thecreditors in the event of a default is again exogenous and is just equal to value of the firmadjusted for bankruptcy costs.Anderson and Sundaresan (1996) and Mella-Barral and Perraudin (1997) also consider

endogenous bankruptcy and they model the strategic behaviour of debtors. In theirmodels, the debtors act strategically and always try to pay as low a coupon as possible.In good times when the liquidation value of the firm is high, the debtors will not paylower than the contracted amount as they would realise that it would then be in thecreditors’ interest to reject their offer and liquidate the firm. However, the debtors mightunderperform the debt contract even if the firm is not experiencing any liquidity problems.They will do this when the liquidation value of the firm is not sufficiently high and thuswhen subsequently it would be not in the creditors’ interest to reject the offer. Thus intheir models, the debtors might default continuously and they will continue to do so untilthe creditors finally reject the offer. At this point the firm will be liquidated.Thus their models are similar to our model in the sense that endogenous bankruptcy

occurs and that this bankruptcy point will in general be different from the default point.However there are a number of differences. We do not consider strategic debt service.In our model the debtors do not act strategically because once a default occurs, debtcovenants are triggered off and this gives the creditors an option to either liquidate thefirm or keep the firm running. In either case the debtors then only get the value of theiroutside option. Thus as long as the firm is in default the debtors’ payoff is limited to theiroutside option and the creditors get most of the bargaining power as they can alwaysthreaten to liquidate. However the creditors are aware of the fact that they cannot runthe firm without the manager because of her inalienable human capital. We thereforeexplicitly model the manager’s inalienable human capital. Subsequently, if the creditorsdecide to keep the firm alive, they have to ensure that the manager stays in the firm. Theycan do this only if they offer the manager at least the value of her outside option. Hencein the renegotiation game of our model, the creditors have bargaining power because theyhold an option to liquidate once the firm enters a default while the debtors’ bargainingpower stems from their inalienable human capital.One important difference between Anderson and Sundaresan (1996) and our model is

that unlike their model, the creditors in our model act strategically once a default occurs.Thus if they decide not to liquidate then they will just pay enough to the manager so asto retain her human capital and will thus in effect become the residual claimants until thefirm exits default.Mella-Barral (1999) obtains some results which are quite similar to ours. Like our

model, one of the objectives of his model is to explain why default rarely coincides withliquidation. Like us, he also shows why it might be rational for the creditors not to liqui-date even if the debtors do not have any bargaining power. This is because renegotiatingthe debt can actually increase the market value of debt by avoiding ill-timed liquidation.However, there are a number of important differences. Mella-Barral considers two

mirror games where either the creditors are in a position to make take-it-or leave-it offers todebtors or the debtors can have the first mover advantage. In the former case the debtorsdo not have any bargaining power and once a default occurs, the creditors themselvesmake self imposed concessions. The concessions are just enough to induce the debtors

The Valuation of Corporate Debt with Default Risk 3

not to default if the firm was being liquidated at an inefficiently early stage.1 Thus in hismodel the debtors do not have any bargaining power if the creditors have the first moveradvantage. On the other hand in our model, the debtors always have bargaining powerover the creditors as they have perfectly inalienable human capital. They can alwaysthreaten (the creditors) to leave the firm if they do not get at least the value of theiroutside option. Furthermore, in our case liquidity problems determine the default point.Conversely, Mella-Barral assumes that liquidity problems do not have any influence on thedefault point as the debtors can keep on issuing equity to avoid a default. Subsequentlyin his model, default is endogenous and is the point where it is optimal for the debtorsto irreversibly exchange their current claim for a residual claim which they will get onbankruptcy. In our set-up default occurs because of liquidity problems and is thereforeexogenous. Further, in our case there is no irreversible exchange of claims and eitherparty may get their original claims if the firm manages to exit default. In Mella-Barral’sanalysis, a capital structure irrelevance result holds after renegotiation. However, theModigliani Miller Theorem will in general not hold in our setting.Perhaps the most significant difference between the two analyses is that in Mella-

Barral’s model once a default occurs it is necessary to offer new debt contracts to investorsin order to exit the default state. The alterations in the debt contract are not temporary.If a default problem arises again, then another debt contract has to be drawn up. ThusMella-Barral allows for an unlimited sequence of new contractual arrangements. Thus inhis analysis new debt contracts have to be drawn up continuously every time the firmenters default. This is not the case in our framework. In our model, if the firm goesinto financial distress and thus defaults, then the firm undergoes a reorganization periodwhere a trustee or an administrator tries to solve the firm’s financial problems. If theymanage to bring the firm out of default then the old contract governs again. If however,the firm’s financial position worsens then finally the firm will be liquidated. Hence anyalterations in the payoff functions of both the debtors and the creditors on default areonly temporary and are thus reversible. This is consistent with the US Trust IndentureAct of 1939 which prohibits firms from permanently changing the ‘core’ terms of the bondindenture, which include the principal amount, the interest rate and the stated maturity,unless all the creditors agree unanimously.2

Recently a number of other studies have also incorporated endogenous liquidation.Leland (1998) extends his earlier work by considering the effects of asset substitution ondebt pricing. As in his earlier work he does not make the separation between defaultand liquidation. Hege and Mella-Barral (2000) extend Mella-Barral (1999) by taking intoaccount multiple creditors. Christensen et al. (2001) consider callable perpetual debtwhere both the upper and lower reorganization points are derived endogenously. Theirwork is an extension of the strategic debt service models of Anderson and Sundaresan(1996) and Mella-Barral and Perraudin (1997) to a set-up where the entire debt ratherthan just the current coupon payment is renegotiated. This feature is similar to ourmodel, as in our framework all the coupon payments, rather than just the current one,are altered in the default region. However, unlike Christensen et al. this alteration is notpermanent and is reversible.

1Mella-Barral (1999) makes a distinction between ‘early’ liquidation and ‘late’ liquidation. In his model‘early’ liquidation occurs if the debtors decide to default too early relative to the first best liquidationpoint. ‘Late’ default occurs if the debtors do not default even if it is (first best) efficient to do so.

2 If the firm goes to Chapter 11, then these core terms can be altered if a two-thirds majority by valueand a simple majority by number is reached within each class of creditors. However unanimity is requiredoutside of Chapter 11.

The Valuation of Corporate Debt with Default Risk 4

With respect to the debt pricing literature our main contribution is to provide a clearseparation between default and liquidation which is consistent with existing bankruptcyregimes. In much of the current debt pricing literature, the entrepreneur maximising thevalue of equity makes the liquidation decision. However the separation of default andliquidation in our model enables us to delegate the liquidation decision to the creditors.Thus in our model default is triggered by the manager, whilst the liquidation decision ismade by the creditors. This seems more natural as the occurrence of default triggers debtcovenants, which in turn might lead to liquidation by creditors.We construct a model where default in general would not lead to immediate liquidation.

In our model, once a default occurs the debtors lose most of their control rights and thecreditors in effect become the residual claimants as long as the firm remains in financialdistress. On the occurrence of a default the control rights are passed to a creditors’representative who then decides whether to continue or to liquidate. This decision willbe in the best interest of the creditors. The firm will be liquidated by the representativecreditor only if from the creditors’ point of view there is no benefit in keeping the firmalive. This will be the case if the firm’s financial position is very dismal or if the firm’sinsolvency further aggravates after default. However, as long as the firm is not liquidatedthe debtors preserve control of the management of the firm because of their inalienablehuman capital. If the firm manages to exit default then the old contract governs againand the debtors then retain all of their control rights and are once again the residualclaimants.This feature of our model, whereby the creditors on default appoint a representative

to make optimal decisions on their behalf but where the debtors still make the day today management decisions of the firm is a good approximation to most of the existingbankruptcy regimes. For instance in the US under Chapter 11, the debtors maintain theircontrol of the management of the firm as the debtor-in-possession or are at least aided bya trustee appointed by the court. Thus even after default the value of equity in practiceis not zero. This is consistent with our model where the value of equity falls to zero onlywhen the firm is finally liquidated. However once the firm files under Chapter 11 themanagement is subject to detailed supervision by the court.In the UK most banks hold a floating charge over the assets of a company and if

the firm is unable to meet its obligations then the banks have the power to appoint anadministrative receiver who then supervises the running of the firm and who has thepower to put the company into liquidation. Alternatively, the creditors in UK can resortto formal reorganization by going to court. The court then appoints an administratorwho has all the powers vested in the board of directors.3 Administration is succeededby liquidation only if the company is unable to survive as a solvent going concern. TheUK Bankruptcy Code closely resembles the South African “judicial management” andthe Australian “official management” and gives relatively more powers to the creditorsvis-a-vis the US Chapter 11.4

In France, the judicial arrangement (Redressment Judiciare) consists of two stages;the observation stage and the execution stage. The observation stage in terms of ourmodel can be interpreted as the reorganization stage whereby both parties try to reachan agreement to keep the firm alive. The execution stage is the liquidation stage and

3Section 17 of the UK Insolvency Act 1986 provides that the administrator shall on his appointmenttake control of all the property to which the company appears to be entitled until the firm is eitherrestored to good health or is liquidated.

4However after the 1994 reform, Chapter 11 returned some powers to the creditors.

The Valuation of Corporate Debt with Default Risk 5

it occurs when reorganization fails. When a firm defaults and goes to court, then thecourt appoints an administrator, a creditor’s representative and a supervisory judge. Thecontrol rights pass from the debtors to the administrator working with the judge. Howeverthe management retains control of the running of the firm. In Germany again as long asthe firm is not liquidated, the debtors retain control of management but are supervisedby an administrator representing the creditors. Thus the structure of our model closelyresembles most of the existing bankruptcy regimes.Like Hart and Moore (1994) a central feature of our model is the inalienability of

human capital. The bargaining power of the manager in our model comes from her abilityto threaten the creditors to repudiate the contract by withdrawing her human capital.However one of the assumptions underlying Hart and Moore’s theory of debt is thatthe manager does not have any outside options and hence has a zero outside wage. Onthe contrary, we do not make that assumption and we assume that the manager alwayshas a valuable outside assumption. This is quite realistic and as argued by Rajan andZingales (2000) with the increase in competition physical assets have become less uniqueand managers today have many outside options.Rajan and Zingales (2000) in their article, “The Governance of the New Enterprise” ar-

gue that powerful forces are changing the nature of the firm and the increased importanceof human capital has led to the breakdown of the traditional vertical integrated firm.Rajan and Zingales (2000) reflecting on the changing nature of the modern enterpriseargue:

...perhaps the most significant change has been to human capital. Recentchanges in the nature of the organisations, the extent and requirements ofmarkets, and the availability of financing have made specialised human capitalmuch more important, and also much more mobile. But human capital isinalienable, and power over it has to be obtained through mechanisms otherthan ownership.

Our debt pricing model also has interesting implications for corporate finance as dis-cussed in Section 3 of the paper. With respect to the corporate finance literature our maincontribution is to show that the Irrelevance Theorem will not hold even in the absenceof informational asymmetries, coordination problems, financial distress costs and taxes.We identify the first best liquidation point and show that the introduction of debt in thecapital structure is conducive to an agency problem whereby creditors liquidate the firmprematurely.The rest of the paper is organised as follows. Section 2 constructs a continuous time

pricing model of the levered firm. We model the reorganization process and obtain an-alytical solutions for equity and corporate debt in terms of the endogenous bankruptcypoint. We show that the default trigger will in general be different from the point ofliquidation. In Section 3 we value the unlevered firm in the same set-up and show that ingeneral capital structure irrelevance will not be obtained. We then characterize the firstbest liquidation point and offer an explanation as to why the Irrelevance Theorem breaksdown. Section 4 provides a discussion of our results. Finally, Section 5 concludes.

2. The Model2.1. The Basic Set-up. We model a firm which is run by one owner-manager en-dowed with specialised human capital. This human capital can be interpreted as atechnical skill which is valuable to the firm and without which the firm cannot be run.

The Valuation of Corporate Debt with Default Risk 6

Alternatively the owner-manager can be thought of as possessing an essential asset whichis vital for the operations of the firm. The manager has many outside options and herreservation income is a. The value of her outside option, a, can therefore be consideredas her income if she decides to move back to her old job. Further, the manager requiresa minimum wage of at least a and she will leave the firm at any period if her income isless than the sustenance level a. Hence the manager will consume at least a per periodbut will consume more if the cash flow of the firm is sufficiently high given that the ownerwill have residual claim in these good states of the world.The owner has limited wealth and therefore requires financing for the firm’s project.

This financing is arranged through the issue of a debt contract.5 We assume that theowner issues perpetual debt to finance the project. The terms of the debt contract requirethat a coupon payment, b, be paid to creditors per instant of time. We do not allow debtservice to be funded through the issue of new securities or through asset sales. To abstractfrom any coordination problems we assume that there exists a representative creditor whomakes optimal decisions on behalf of a cohesive group of creditors.6

We suppose that capital markets are frictionless and that there are no informationalasymmetries between agents. Further, all agents are risk neutral and the term structureof interest rates is flat with a nonstochastic rate, r. The assumption of risk neutrality iswithout much loss of generality. The model can be developed using risk neutrality withordinary probabilities, or alternatively under risk aversion with risk neutral probabilities.(See Harrison and Kreps (1979) for a discussion.) Finally, we assume that the liquidationvalue of the firm’s project is given by K.

2.2. The critical default trigger. The firm’s underlying state variable is cash flows,x, which follow a geometric brownian motion, i.e.

dx = αxdt+ σxdw (1)

where the parameters α and σ represent the drift and volatility terms respectively anddw is the increment of a standard Wiener process. Note that we do not assume that thereexist traded securities such that any new claim can be priced by dynamically replicatingalready existing securities. Hence we do not require the completeness of markets and thestochastic changes in x need not be spanned by existing assets in the economy. Manycorporate finance valuation models use the classical contingent claims pricing approachand use the value of the unlevered firm as the basic state variable. However contingentclaims analysis assumes that markets are sufficiently complete and as pointed out byChristensen et al. (2001) it is impossible to have both the unlevered firm and the optimallylevered firm to coexist as trading assets.In our model, default occurs because of liquidity problems. This is because like Ander-

son and Sundaresan (1996) we assume that debt service is met out of cash flows and thatthe firm cannot issue additional equity or debt to avoid a default. This is not a very strin-gent assumption as it might first appear. In practice, debt covenants frequently restrictthe issue of additional debt with senior or equal status. Similarly, loan indentures quite

5The debt contract is thus justifiable given the limited wealth of the owner.6This is a reasonable assumption which is consistent with most of the exsiting bankruptcy codes. For

instance, Chapter 11 provides an automatic stay against creditors’ claims to avoid credior harrassmentduring the reorganization process. Thus creditors are prevented from foreclosing on their collateral.Nevertheless coordination problems can exist during the voting process (unless a ‘cram down’ is imposed)or in out-of-court restructurings.

The Valuation of Corporate Debt with Default Risk 7

often forbid the liquidation of firm’s assets by owners as this could potentially underminecollateral values.To simplify our analysis we also do not allow the firm to issue junior debt or equity. In

some respects this brings us closer to reality. It is very rare for firms in financial distressto issue equity so as to avoid a default on its financial obligations. In a rational market,agents would realise that equity is being issued to prevent a default and thus any saleof equity would further decrease the suppressed value of shares.7 Firms can in principleissue junior debt to avoid a default but this would be very costly for the issuing firm.The reason is that if a firm is in financial distress then the addition of further debt in itsfinancial structure would just increase the debt servicing obligations of the firm and henceincrease the likelihood of a default in future periods. Further, as argued by Anderson andSundaresan (1996) the fixed costs relating to the issue of securities are likely to be higherfor firms in financial distress and hence it may not be feasible for the firm to issue newsecurities. For all these reasons the addition of new claimants would most likely just delaydefault for a short period of time rather than prevent it. Nevertheless extending the modelto allow for the issue of new securities would be an interesting area for further research.Since in our model, default occurs because of liquidity problems facing the firm, hence

the critical default point is determined exogenously. Default occurs whenever the firmscurrent cash flows are not enough to meet its debt obligations. We conjecture the following.

Conjecture 1. Assume that the firm has issued perpetual debt with coupon payment band principle b/r.8 Suppose that the value of the manager’s outside option is given by

a. Let∧x denote the critical level of cash flows below which the firm will default on its

debt payments and suppose that the firm cannot issue new securities or liquidate assetsto service its debt. Then the critical default point of the firm is given by

∧x = a+ b (2)

It is quite straightforward to see why Conjecture 1 holds if we suppose for the momentthat the debtors will not act strategically when servicing their debt payments. Clearly ifdebtors do not act strategically then the default point will be given by Eqn. (2). Thisis because the owner needs to consume at least a per period. Thus in every period theowner will first take out a portion a of the cash flows before servicing its debt. Since theowner is obliged to pay a coupon of b period to the bondholders, thus default will occurwhenever the cash flow falls below a+ b. Thus the critical default point is

∧x.

Once a default occurs, debt covenants are triggered which gives creditors the right toliquidate. However the creditors might not find it in their interest to liquidate the firmand hence might want the firm to continue. Hence the occurrence of a default gives thecreditors an option to liquidate which the creditors may or may not exercise.A key feature of our model is that if the creditors decide to continue then the control

rights are essentially transferred to the creditors as long as the firm is in the default state.For our purposes, control rights can be defined as the right to control the distribution

7This would also be true in the presence of asymmetric information in financial markets. Myers andMajluf (1984) argue that the issue of equity is taken as a bad signal by market participants and thistherefore depresses stock prices.

8Like Mella-Barral and Perraudin (1997) we assume that the debt principal is b/r. This is purely forexpositional purposes and is without much loss of generality.

The Valuation of Corporate Debt with Default Risk 8

of the firm’s resources.9 The reason why a transfer of control rights occurs on default isthat as long as the firm is in default the creditors are getting less than their contractedpayments and hence have the first right to the firms cash flows.10 We claim the following.

Claim 1. The debtors will not default unless faced by liquidity problems and hence willnot act strategically if a transfer of control rights occurs from the debtors to the creditorsonce the firm enters default. Hence, Conjecture 1 is true.

To show why the owner will not act strategically, suppose on the contrary that theowner decides to default even if the firm is in a healthy state and hence can meet its debtobligations. Suppose the owner decides to underperform the contract and pays less thanthe contracted coupon payment, b, to the creditors and further threatens to withdrawher inalienable human capital if the creditors do not accept her offer. If the owner paysan amount slightly over rK to the creditors, then the creditors would not find it in theirinterest to liquidate as liquidation will just fetch them a scrap value of K.11 However, thecreditors can decide not to exercise their option to liquidate and would actually prefer tolet the firm continue if its fundamentals are strong enough. If the creditors continue thenthey will realise that the owner’s threat to withdraw her human capital is not credible,given that they would now have the control rights. Once the creditors get the controlrights they will offer the owner an amount slightly over a such that the owner will thenhave to accept the creditors’ offer. The owner will foresee this outcome ex ante and willrealise that her threat to withdraw her human capital is not credible. Hence the subgameperfect strategy of the owner will be not to default unless faced by liquidity problems.Thus Conjecture 1 is consistent with a subgame perfect Nash equilibrium.A key distinction between our model and the strategic debt servicing models of An-

derson and Sundaresan (1996) and Mella-Barral and Perraudin (1997) is that in our casea transfer of control rights occurs from the debtors to the creditors as long as the firmremains in default. It is because of this shift of the control rights that the debtors in ourset-up do not act strategically. They are aware of the fact that on default, they will losetheir control rights and hence will then be limited to the value of their outside option.Hence the debtors do not find it advantageous to strategically service the debt. Howeverif the debtors are always in charge of the control rights of the firm then they will find itworthwhile to act strategically as in that case they can afford to make a take-it-or-leave-itoffer and the creditors will not be able to respond with an offer of their own.

2.3. The reorganization process. Once the cash flows fall below the critical level∧x, a default occurs and debt covenants are triggered. The creditors then have an optionto either let the firm continue or to stop and liquidate. We thus solve for the creditors’optimal stopping problem. Before solving for the creditors’ optimal stopping rule we needto consider the game between the creditors and the manager on the occurrence of a default.We thus embed a game theoretic approach in our continuous time asset pricing model toascertain the equilibrium payoffs in different states of the world. The outcome of these

9Strictly speaking a distinction should be made between control rights and cash flow rights. The latterwould be the right over the firm’s cash flows whilst the former would be the right to control the liquidationdecision. This difference does not matter in our case as creditors have both control rights and cash flowrights when the firm is in default.10This can be interpreted as the appointment of a creditors’ respresentative to oversee management

and to ensure the protection of creditors’ interest.11rK will be less than b if the debt is risky.

The Valuation of Corporate Debt with Default Risk 9

games critically depend on the relative bargaining power between the two parties. Unlikemost of the debt pricing models which give all the bargaining power to either the creditorsor the debtors, we consider a case where both parties have some bargaining power duringthe reorganization process.As argued by Rajan and Zingales (2000), there are different sources of power. Power

can be derived through the rules of the bargaining process. For instance, the party withthe first mover advantage might exert some power on the other. Alternatively, a negotiatorcan exert power if she possesses some valuable resource which is required in the productionprocess. These are the two sources of power that we consider in our game.In the game that we consider, creditors act as the Stackelberg leader and make take-

it-or-leave-it offers to managers. Once a default is triggered, creditors can either continueor liquidate the firm. If they decide to continue than they get the control rights. Theythen realise that if they pay anything less than a to the owner-manager then the managerwill take her human capital and leave. Further they are aware that the manager’s humancapital is inalienable and the firm cannot be run without her specialised knowledge. Hencethe creditors pay the manager just enough so as to retain her human capital. They dothis by offering the manager the value of her outside option. The manager hence acceptsthe offer. Thus rejecting never occurs in equilibrium. Hence in the subgame perfectequilibrium, the creditors offer an amount a in every period to the manager which themanager accepts. This will be the case as long as the firm is in default.Note that the creditors derive their bargaining power from their option to liquidate

the firm. It is this option which gives them the first mover advantage. The manager onthe other hand influences the bargaining process because of her inalienable human capital.The higher the value of her outside option the more will be her share of the bargainingpower. However if the value of her outside option is too high, then the creditors might bebetter off by just liquidating the firm.Once a default occurs, a fraction ϕ of the cash flows is lost every instant of time. This

fraction represents the cost of financial distress, for instance, lawyers’ charges, adminis-trator’s fees, or possibly a loss of possible profitable investment opportunities as managersmight be distracted because of their dealings with the creditors.Hence if the creditors continue then their payoff will be (θx− a) per period, where

θ = 1 − ϕ. If they liquidate then they get K. If the creditors decide to continue, thenthey keep their option alive and in the next time period (if the firm stays in default) theyagain have the option to liquidate. Thus, there is value in waiting. Hence by solvingthe creditors’ dynamic programming problem, we can determine the value of their optionwhich is given by

F (x) = max {K, [θx− a] dt+ E [F (x+ dx) exp(−rdt)]} . (3)

where E is the expectations operator. Note that the continuation value is given by C(x) =[θx− a]dt+E [F (x+ dx) exp(−rdt)]. It is just the payoff which the creditors appropriatein the current period plus the expected present value of the option which they retain.The liquidation point, x∗, is thus determined endogenously. It is the point at which thecreditors are indifferent between continuing and liquidating. Hence it is the value of xsuch that the continuation value just equals the liquidation value. When x < x∗, theoptimal decision for the creditors is to liquidate.As shown in Figure 1, the default point differs from the liquidation point. When x

falls below∧x, default occurs but this need not automatically lead to liquidation. If the

firm’s condition further deteriorates and its liquidity problems are aggravated then the

The Valuation of Corporate Debt with Default Risk 10

x* xDefault region x̂

Figure 1: Default and Liquidation. The figure shows how default and liquidation are determinedby the state variable. Default occurs when the state variable hits

∧x. Liquidation occurs when

the state variable falls further to x∗.

firm would finally be liquidated by the creditors. Hence liquidation will take place whenthe state variable falls below x∗.

2.4. Valuation of equity and corporate debt. As long as the firm is in the defaultregion, as depicted in Figure 1, the creditors will have the control rights. However, if thefirm manages to overcome its liquidity problems and is restored to a healthy condition,then the control rights are transferred back to the debtors. Thus the creditors are in effectthe residual claimants as long as the firm stays in default. Nevertheless as long as thefirm is not liquidated there is always the likelihood that the debtors will retain the controlrights. Thus for low firm values the creditors act as the residual claimants but for highenough firm values the debtors will be the residual claimants.Since the control rights can alter from the debtors to the creditors and vice versa, thus

the payoffs of both the parties will be state dependent. Furthermore any change in thepayoff function will be temporary and reversible until and unless the firm is liquidated.This feature of our model brings us closer to reality and we do not need to rely on payofffunctions, the changes of which are permanent and hence irreversible.As long as the firm is doing well and is not in default, the equityholders will be the

residual claimants and their payoffs will be given by x − a − b. However in bad statesof the word, the equityholders will just earn the value of their outside option and hencetheir payoff will be zero.12 The payoff to the equityholders is thus given by

e(x) =

(x− a− b if x ≥ ∧

x

0 if x <∧x. (4)

Therefore, at any instant the profit flow to the equity holders is given by

e(x) = max [x− a− b, 0] .

Notice that the equityholders have a call option on the firm at every instant in time.The option, if exercised at time tmeans that the equityholders will appropriate the currentearnings of the firm in time t given an exercise price of a+b. Since each option can only beexercised at that very instant, the equityholders have a number of European call options.However once the state variable hits x∗, the call option of the equityholders becomesworthless as the firm is then liquidated. Hence the value of equity is the sum of the12Note that the value of the owner-manager’s outside option is correctly treated as an opportunity cost

for the firm.

The Valuation of Corporate Debt with Default Risk 11

value of all these call options and it is therefore dependent on the state variable. As xapproaches x∗, the value of equity declines. Conversely, the value of equity increases asthe earnings of the firm rise.In this model, equity and debt are time homogenous securities and hence their values

are time independent. Using standard no-arbitrage arguments the values of equity anddebt can be expressed in terms of the cash flow process. It can be shown that, in thisenvironment, the value of any time independent claim which is a function of the cash flowprocess, x, will satisfy the following ordinary differential equation

1

2σ2x2S00(x) + αxS0(x)− rS(x) + s(x) = 0 (5)

where S(x) is the value of security S which is a twice continuously differentiable functionof the state variable x, and s(x) is the payoff function of that security.Thus the value of equity, E, will satisfy the following differential equation

1

2σ2x2E00(x) + αxE0(x)− rE(x) + e(x) = 0 (6)

where e(x) is as defined in Eqn. (4). To solve for the value of equity we need appropriate

boundary equations. Let E1 be the value of equity when x ≥ ∧x and E2 be the value of

equity when x <∧x. Then the value of equity will obey the following boundary conditions

limx→∞E1(x) =

x

r − α −a

r− b

r(7)

E1

³∧x´

= E2

³∧x´

(8)

E01³∧x´

= E02³∧x´

(9)

E2 (x∗) = 0. (10)

Eqn. (7) will hold if asset prices are free of speculative bubbles. It just says that asx grows large the value of equity approaches the expected discounted integral of futurepayoffs which will accrue to the equityholders, i.e. as x increases the value of equityapproaches Et

R∞t

(x−a−b) exp[−r(u−t)]du. Eqn. (8) is the value-matching condition atdefault. Eqn. (9) says that the slope of the value function of equity has to be continuousat the default point. This follows from Claim (1) as it is optimal for debtors to defaultonly when the firm is facing liquidity problems. Finally, Eqn. (10) is the value-matchingcondition at liquidation and it says that the value of equity finally falls to zero when thefirm is liquidated.Solving the differential equation (6), subject to the above boundary conditions, we

show in the Appendix that we get the following result.

Proposition 1. Let E(x) denote the value of the levered firm’s equity. Assume that thefirm has issued perpetual debt with coupon payment b and principle b/r. Suppose thatthe value of the manager’s outside option is given by a. Then if the debt is risky13 , i.e.K < b/r, the value of equity is given by13For completeness, in the appendix, we also derive the value of equity for the simpler and less interesting

case, when debt is riskless, i.e. when K ≥ b/r.

The Valuation of Corporate Debt with Default Risk 12

E(x) =

0 if x < x∗

1∧xξ1−∧

xξ2x∗ξ1−ξ2

·nHJ

³ ∧x

r−α −∧xr

´− 1

J(r−α)

o ∧xξ2

+∧x

r−α −∧xr

¸× £xξ1 − x∗ξ1−ξ2xξ2

¤if x∗ ≤ x < ∧

xx

r−α − ar − b

r +hHJ

³ ∧x

r−α −∧xr

´− 1

J(r−α)

ixξ2 if x ≥ ∧

x

(11)

where H and J are constants with the following values

H =ξ1∧xξ1−1 − ξ2x

∗ξ1−ξ2∧xξ2−1

∧xξ1 − ∧

xξ2

x∗ξ1−ξ2

J = ξ2∧xξ2−1 −H∧

xξ2

The powers ξ1 and ξ2 are the positive and negative roots respectively of the characteristicquadratic equation ξ (ξ − 1)σ2/2 + ξα− r = 0.

The solution of the equity function from Proposition 1 is illustrated in Figure 2. Notethat the value of equity is a convex function above the default point but is a concavefunction in the default region. This is because equityholders are the residual claimantsonly in the good states of the world. Further also note that the value of equity is positiveeven in the default region even though the economic payoff to the equityholders in thisregion is zero. This is because as long as the firm is not liquidated, there is always apositive probability that the firm might exit default and hence the equity holders mightbe able to earn economic profits in the future.Next we value the corporate debt issued by the firm. The payoff function of bond-

holders holding risky debt is given by

p(x) =

b for x ∈ [

∧x,∞)

θx− a for x ∈ [x∗,∧x)

rK for x ∈ [0, x∗)(12)

where x∗ is the endogenous point of liquidation chosen by the bondholders. Eqn. (12)states that the bondholders get the full contracted payment in good states of the world.However, in the default region, the bondholders get less than their contracted couponpayment because of the insufficient cash flows generated by the firm. The payoff of thebondholders in the default region is θx − a per instant of time, where θ = 1 − ϕ. Thisis because, as long as the creditors decide to keep the firm alive, they have to pay themanager the value of her outside option, a. Furthermore, in this state of the world,a fraction ϕ of the firm’s cash flows is lost due to financial distress costs. Opler andTitman (1994) find evidence that levered firms incur substantial financial distress costsin industry downturns. These losses might take different forms. They might be customerdriven as customers might be reluctant to do business with financially distressed firms.These losses might also be competitor driven if it is the case that financially stronger firmstake advantage of their weaker counterparts by aggressive pricing in an attempt to driveout the firms experiencing distressed times. More directly, these costs will also reflect

The Valuation of Corporate Debt with Default Risk 13

lawyers fees, etc.14 Finally, the bondholders appropriate the scrap value, K, when thefirm is liquidated and their payoff henceforth is rK per instant of time.The bondholders’ payoff function is piecemeal right continuous and is depicted in panel

(a) of Figure 3. Note that it is discontinuous at the points∧x and x∗. In the absence of

arbitrage, the value function of debt, B(x) and its first derivative B0(x) must both be

continuous at the points∧x and x∗. Thus, the payoff function will be discontinuous if and

only if the second derivative of the value function, B00(x) is discontinuous at the points∧x

and x∗. (This is clear on examination of the ODE (13)). Dumas (1991) has shown that foroptimal stopping problems, smooth-pasting remains a condition only involving the firstderivative of claim values and not the second derivative. This therefore would explainthe discontinuities in the payoff function.15 The payoff function of the equityholders (4)analogous to the payoff function of the bondholders will be piecemeal right continuousand for the same reasoning will be discontinuous at the point

∧x.

The value of debt, B(x), will satisfy the following differential equation

1

2σ2x2B00(x) + αxB0(x)− rB(x) + p(x) = 0 (13)

where p(x) is as defined in Eqn. (12). To solve for the value of corporate bonds we need

to specify appropriate boundary conditions. Let B1 be the value of debt when x ≥ ∧x. We

know that once a default occurs, the bondholders have the option to liquidate, and thevalue of their option is given by F (x) = max {K,C(x)}, where C(x) as defined earlier is thevalue of debt in the continuation region, i.e. C(x) = (θx− a) dt+E [F (x+ dx) exp(−rdt)].Then the value of corporate debt will satisfy the differential equation (13) subject to thefollowing boundary conditions

limx→∞B1(x) =

b

r(14)

B1

³∧x´

= C³∧x´

(15)

B01³∧x´

= C0³∧x´

(16)

C (x∗) = K (17)

C0 (x∗) = 0. (18)

These boundary conditions have the following interpretations. Eqn. (14) is the no bubblescondition which states that as the state variable approaches higher and higher values, thevalue of debt approaches the value of risk free debt. In the limit, when the cash flow hasan infinite value, the debt of the firm is essentially risk free and its value is just equalto the present value of the coupon payments. Eqn. (15) is the value-matching conditionat the default point. Eqn. (16) states that the value of debt must be continuously14The study by Opler and Titman (1994) finds that the costs of financial distress are more pronounced

for highly leveraged firms. Thus, we would expect ϕ to vary with leverage. It will be relatively higher forfirms with more debt in their capital structure.15Thus the “super-contact condition” whereby the smooth-pasting condition can be expressed in terms

of the second derivative will not apply to our case. See Dumas (1991) for details.

The Valuation of Corporate Debt with Default Risk 14

differentiable across x. Since the Brownian motion of the state variable can diffuse freelyacross the default boundary, for no arbitrage the claim value cannot change abruptly.Dixit (1993) shows that the if the flow payoff function changes, then the value functionshould not change abruptly and thus the first derivative of the value function should becontinuous for the absence of arbitrage. (See Karatzas and Shreve (1998) for a morerigorous discussion). Eqn. (17) is the value-matching condition at liquidation. From theBellman equation (3) we know that in the creditors’ stopping region, the value of theiroption to liquidate is just equal to the scrap value of the firm, and hence by continuity wecan impose the value-matching condition (17). Eqn. (18) is the standard smooth-pastingcondition which determines the optimal stopping point for the creditors. Thus when thestate variable falls below x∗ the creditors will find it in their interest to enforce liquidation.As shown in the Appendix, solving the differential equation (13) given the boundary

conditions just described yields the following result.

Proposition 2. Assume that the firm has issued perpetual debt with coupon payment band principle b/r. Suppose that the value of the manager’s outside option is given by a.Suppose that a fraction ϕ of the firm’s cash flows is lost in financial distress costs as longas the firm is in default and that θ = 1−ϕ. Let B(x) denote the value of corporate debt.Then if the debt is risky, i.e. K < b/r, the value of debt is given by

B(x) =

K if x < x∗θxr−α − a

r +h¡ θ(ξ2−1)

(ξ1−ξ2)(r−α)

¢x∗1−ξ1 − ξ2

ξ1−ξ2

£K + a

r

¤x∗−ξ1

ixξ1

+h¡ 1−ξ1

ξ1−ξ2

¢θx∗1−ξ2

r−α + ξ1

ξ1−ξ2

£K + a

r

¤x∗−ξ2

ixξ2

if x∗ ≤ x < ∧x

br + {

³1

r−α − 1r

´ ∧x

1−ξ2

+h¡ θ(ξ2−1)

(ξ1−ξ2)(r−α)

¢x∗1−ξ1 − ξ2

ξ1−ξ2

£K + a

r

¤x∗−ξ1

i ∧xξ1−ξ2

+h

1−ξ1

ξ1−ξ2

θx∗1−ξ2

r−α + ξ1

ξ1−ξ2

¡K + a

r

¢x∗−ξ2

i}xξ2

if x ≥ ∧x

.

(19)If the debt is riskless, i.e. K ≥ b/r, then the value of debt is given by B(x) = b/r.

While we have derived a closed form solution for corporate debt, the endogenous liqui-dation point x∗ cannot be expressed in closed form. Nevertheless, root finding algorithmscan numerically calculate x∗, given the values of other parameters. Given Proposition 2,the Appendix shows that x∗ will be the solution to Eqn. (16). Thus the threshold x∗ canbe computed numerically from Eqn. (16) given other parametric values.Given the form of the solution, it might be the case that for a certain range of para-

metric values x∗ < a. If that happens to be the case, then it would actually imply thatthe bondholders might be willing to inject cash into the firm if the state variable dropsbelow a/θ. (This can readily be seen from inspection of the payoff function in Eqn. (12)).This might be the case if the scrap value of the firm is low enough relative to a high valueof the manager’s outside option. Intuition would then suggest that if the manager has alot of bargaining power during the reorganization process, bondholders might be willingto inject cash so as to retain the manager’s human capital as they would realise thatliquidation would fetch them a very low value. Nevertheless they will ultimately liquidateif the gap between the state variable, x, and a widens.

The Valuation of Corporate Debt with Default Risk 15

Cash flow (x)

K

x*

b/r

Values

E(x)

B(x)VL(x)

Liquidation Defaultx̂

Figure 2: The dynamics of the firm and its securities. The figure shows the value function ofthe levered firm, VL(x), and the values of the firm’s equity, E(x), and debt, B(x), as functionsof cash flow, x.

The solution from Proposition 2 is depicted in Figure 2. Note that as is the usual case,the value of debt is a concave function of the state variable in good states of the world.However, in the default region the value of debt is a convex function of the state variablereflecting the fact that the bondholders are in effect the residual claimants as long as thefirm remains in default. They cannot recoup the full contracted coupon payment butnevertheless extract as much as is possible after paying off the manager her value of theoutside option.The figure illustrates that default occurs when the state variable first hits

∧x. However

this point might not automatically result in liquidation as creditors might find it in theiradvantage to retain their option to liquidate. Nevertheless if the financial position of thefirm is further aggravated, the creditors might find it in their interest to liquidate thefirm. Liquidation will occur when the cash flow falls below the critical level x∗. Thus theliquidation point will in general differ from the point of default.

3. Does the Modigliani-Miller Theorem hold?Modigliani and Miller (1958) showed that the market value of any firm is independentof its capital structure and thus the value of a levered firm should be the same as thevalue of an unlevered firm. However, we show that given our set-up, capital structureirrelevance in general will not be obtained even in the absence of financial distress costsand taxes. We first value an unlevered firm in the same set-up as before and then we

The Valuation of Corporate Debt with Default Risk 16

provide an explanation as to why the value of the unlevered firm would in general differfrom the value of its levered counterpart.

3.1. The Value of an Unlevered firm. We now consider the valuation of a pureequity firm. Apart from no debt in the capital structure, the set-up is essentially the sameas described in Section 2.1. The owner-manager running the firm can always shut downthe firm for a scrap value of K and take her outside option a. Given a competitive labourmarket, a will also be the minimum level of consumption required by the manager everyperiod in time. Thus the owner-manager in effect has a put option to liquidate the firm,where the exercise price of the option is K.The value of the owner’s put option, G(x), is given by

G(x) = max {K, (x− a) dt+ E [F (x+ dx) exp(−rdt)]} . (20)

Hence the manager in every instant of time, has an option to shut down the firm andfetch a value of K or to continue and get a net payoff of x− a in the current period plusretain the option to liquidate in future periods. The point of liquidation is given by

L∗ = max(a, x∗u) (21)

where x∗u is the value of the cash flows such that the scrap value just equals the continua-tion value. The constraint (21) is imposed as the manager needs to consume a minimumamount of a per period.16 Hence the liquidation point will be the higher of the valueof the manager’s outside option and the critical level of the state variable such that thecontinuation value in Eqn. (20) just equals the scrap value.The payoffs of the owner-manager are given by

v(x) =

½x− a if x ≥ L∗rK if x < L∗ (22)

where L∗ is the point of liquidation chosen by the manager as described above.The value of the unlevered firm, Vu, will satisfy the following differential equation

1

2σ2x2V 00u (x) + αxV 0u(x)− rVu(x) + v(x) = 0 (23)

where v(x) is as defined in Eqn. (22). To solve for the value of the unlevered firm,appropriate boundary conditions are required. In the case of the unlevered firm, we justhave three boundary conditions, given that we do not have the issue of default here. Theboundary conditions are as follows

limx→∞Vu(x) =

x

r − α −a

r(24)

Vu(x∗u) = K (25)

V 0u(x∗u) = 0. (26)16Even though the manager will always be better off by stopping at x∗u rather than a, nevertheless if

a > x∗u the manager will have to shut down when the state variable drops below a. This will be the casegiven that the manager has limited wealth and cannot inject further cash into the firm.

The Valuation of Corporate Debt with Default Risk 17

These boundary conditions have straightforward interpretations. Eqn. (24) is the nobubbles condition. Eqns. (25) and (26) are the value-matching and smooth-pastingconditions respectively.We show in the Appendix that solving for the differential equation (23) subject to the

boundary conditions (24),(25) and (26), the following result is obtained.

Proposition 3. Let Vu(x) denote the total value of the unlevered firm. Assume that thevalue of the manager’s outside option is given by a. Then the value of the unlevered firmis given by

Vu(x) =

(K if x < L∗x

r−α − ar +

h¡K + a

r

¢x∗−ξ2 − x∗1−ξ2

r−αixξ2 if x ≥ L∗ (27)

where

L∗ = max(a, x∗u)

and

x∗u =hK +

a

r

iµξ2(r − α)

ξ2 − 1

¶. (28)

Here, ξ2 is the negative root of the characteristic quadratic equation ξ(ξ−1)σ2/2+ξα−r =0.

Note that now we are able to find a closed form expression for the liquidation pointgiven the relatively simple nature of the valuation of the unlevered firm in this set-up.For a certain range of parametric values it will be the case that x∗u < a. In fact, it canbe shown that this will be the case if and only if K < aλ, where the parameter λ is aconstant such that λ ≡ [(ξ2 − 1) /ξ2 (r − α)− 1/r]. The intuition behind this is that ifthe scrap value is very low compared to the opportunity cost of staying in the business,then the manager would actually prefer to inject some money and keep the firm alive aslong as the state variable is above x∗u. However as discussed in footnote 16, the managerhas limited wealth and cannot inject further cash into the firm. Further, the managerconsumes at least a per period. Given this resource constraint, the manager will haveto follow the liquidation rule as defined in Eqn. (21). As discussed in Section 3.2, thiswealth constraint can actually be a potential source of inefficiency for the unlevered firm.Before discussing the efficiency of the unlevered firm vis-a-vis the levered firm, it is

useful to restate the above fact in the following condition.

Condition 1. L∗ = a if and only if K < aλ, where λ ≡ [(ξ2 − 1) /ξ2 (r − α)− 1/r].

Condition 1 states that the unlevered firm will be liquidated at the point where thestate variable hits the value of the manager’s outside option if and only if the scrap valueof the firm is very low relative to the opportunity cost of continuing the firm.

3.2. First Best Liquidation, inefficiencies, and Capital Structure Relevance.We now turn to the interesting issue of whether the value of the unlevered firm is highercompared to its lever counterpart. To address this issue we first need a benchmark forefficiency. The values of the levered and unlevered firms will differ if their liquidation

The Valuation of Corporate Debt with Default Risk 18

points differ. Further, a measure of the inefficiency of the firm would be the magnitudeof the difference of its liquidation point relative to the first best liquidation point.Given our analysis, it is straightforward to identify the first best liquidation point of a

firm. By definition, the first best liquidation point of a firm is the point which maximisesthe value of the firm. For the unlevered firm, the first best liquidation point also maximisesthe value of equity. As discussed in Section 3.1, the value of equity will be maximised if themanager follows the liquidation rule such that she liquidates whenever the cash flows fallbelow x∗u. Nevertheless, as discussed earlier, the manager might have to liquidate earliergiven her wealth constraint, even though she would have been better off if liquidationhad occurred at x∗u. Thus, x∗u is the optimal liquidation point as it is identified by thesmooth-pasting condition, which itself by definition is an optimality condition. Thus weknow that the value of the unlevered firm will be at a maximum if the manager followsthe liquidation rule such that she liquidates when the state variable hits x∗u. We have thusproved the following Proposition.

Proposition 4. The optimal liquidation rule, which maximises the value of the firm, isto liquidate the firm when the state variable falls below x∗u. Thus the first best liquidationpoint is given by

x∗u =hK +

a

r

iµξ2(r − α)

ξ2 − 1

¶Having identified the first best efficient liquidation rule, we next turn to the question

of whether the Modigliani-Miller Theorem holds and if not then whether the value of theunlevered firm is necessarily higher than the value of the unlevered firm. From inspectionof Proposition 3 it is immediately clear that the value of the unlevered firm is not equalto the sum of the value of equity and the value of debt of the levered firm. (The values ofthe levered firm’s equity and debt are given by Propositions 1 and 2 respectively.) This istrue even if the financial distress costs are set to zero. Therefore, an important corollaryto Proposition 3 is

Corollary 1. The Modigliani-Miller Theorem does not hold even in the absence of finan-cial distress costs and taxes.

The reasoning behind the capital structure relevance result obtained is subtle butnaturally follows from our model. Capital structure irrelevance will not be obtained if theliquidation point of the levered firm differs from the liquidation point of the unleveredfirm. For now, lets suppose that Condition 1 is not satisfied and thus x∗u > a. Thus themanager in the unlevered firm follows the first best liquidation rule, and liquidates whenthe state variable hits x∗u as defined in Eqn. (28). Corollary 1 therefore implies that thecreditors’ liquidation point will differ from the liquidation point of the manager in theunlevered firm, i.e. x∗ 6= x∗u. We now provide the intuition behind this.In the levered firm, once a default occurs the creditors have the option to liquidate the

firm. In the unlevered firm, the manager always has an option to liquidate the firm fora scrap value of K. In fact it will be the case that the creditors will exercise their (put)option to liquidate earlier vis-a-vis the manager in the unlevered firm even in the absenceof financial distress costs. Figure 3 compares the payoff function of the creditors andof the manager in the unlevered firm. For simplicity assume that there are no financialdistress costs, i.e. θ = 1. Then note that both the creditors and the manager have thesame payoff function for low fundamental values of the firm. In bad states of the world,

The Valuation of Corporate Debt with Default Risk 19

(x)Cash flow

b

x*

rK

(a)

x̂

Payoffs

(x)Cash flow

rK

Payoffs

(b)

x*u

Figure 3: The payoff functions. Panel (a) of the figure depicts the payoff function of thebondholders in the levered firm while panel (b) depicts the payoff function of the owner managerin the unlevered firm. Both of them have options to liquidate, but in the absence of any wealthconstraints, the bondholders will liquidate earlier given their capped payoff. Because of this“capped payoff” effect, the value of the levered firm will be lower than the value of the unleveredfirm in the absence of any wealth constraints of the manager in the unlevered firm.

the creditors are the residual claimants and their payoff is thus x − a. In the unleveredfirm, the manager always is the residual claimant with a payoff of x − a as long as thefirm is not liquidated. Thus in good states of the world, while the manager is still theresidual claimant and earning x− a, the creditors payoff will be capped by the amount ofthe coupon payment b.Thus both the creditors and the unlevered firm manager have put options to liquidate

but these options have different values as the payoff functions of the two parties differ. Infact the creditors put option is less valuable due to an upper limit to their payoffs. Thecreditors realise that their payoffs are capped by the amount of their coupon paymentand they take this into account when determining the liquidation point. Because of theupper limit to the creditors’ payoff, they would liquidate earlier compared to the managerin the unlevered firm. It will therefore always be the case that x∗ > x∗u. We thus havethe following proposition.

The Valuation of Corporate Debt with Default Risk 20

Proposition 5. The creditors will always liquidate the firm prematurely relative to thefirst best liquidation point.

Thus if Condition 1 is not satisfied it will always be the case that the creditors willliquidate earlier compared to the unlevered manager and thus the value of the unleveredfirm will be higher than the value of the levered firm, given the inefficiency induced inthe levered firm by the creditors’ inefficient liquidation decision. The creditors liquidationdecision would maximise the value of their claim but not the value of the firm as a whole.Note that the Coase theorem will fail in this environment given the limited wealth of themanager. Thus both the parties will be unable to bargain their way towards the efficientoutcome.More generally, depending on the parametric values, the following three cases might

arise.Case 1 : a < x∗u < x∗. In this case, Condition 1 is not satisfied and thus L∗ = x∗u.

Thus now the wealth constraint of the managers in the unlevered firm is not binding andhence they will follow the first best liquidation rule as defined in Proposition 4. But sincex∗u < x∗, the creditors in the levered firm will liquidate too early and thus the value ofthe levered firm will be less than the value of the unlevered firm.

Case 2 : x∗u < a < x∗. Note that now since x∗u < a, Condition 1 holds and thus thewealth constraint of the manager in the unlevered firm is binding. Therefore the managerof the unlevered firm will now liquidate when the state variable hits a and thus givenher resource constraint, the manager will be unable to follow the first best liquidationrule. Nevertheless the value of the unlevered firm will still be higher than the value of thelevered firm given that the liquidation rule followed by the creditors in the levered firmis even more inefficient relative to the liquidation rule being followed by the manager inthe unlevered firm. More Formally, |L∗ − x∗u| < |x∗ − x∗u| and thus the deviation from thefirst best liquidation rule is higher in the case of the levered firm.

Case 3 : x∗u < x∗ < a. Again, as in Case 2, Condition 1 holds and thus the resourceconstraint of the manager in the unlevered firm is binding. Thus the liquidation pointof the unlevered firm’s manager is now given by L∗ = a. However one critical differencebetween Case 1 and Case 2 is that the resource constraint is now binding at a levelwhich is higher than the liquidation point chosen by the creditor in the levered firm. Theimplication is that now |L∗ − x∗u| > |x∗ − x∗u| and hence now the deviation from the firstbest level is actually higher for the unlevered firm. Thus the unlevered firm will be moreinefficient, in terms of its liquidation threshold, relative to the levered firm. Thus, givenCase 3, the value of the levered firm will actually be higher than the value of the unleveredfirm.We have now identified all possible cases that might arise. Our general result is that

the value of the levered firm will in general be not equal to the value of the unleveredfirm. Furthermore, the levered firm would never achieve the first best given an upperlimit to the creditors’ payoff. Nevertheless the unlevered firm might be able to achievethe first best if the wealth constraint of the manager is not binding. However, if the wealthconstraint of the manager is binding, then the value of the unlevered firm may or maynot be higher than the value of the levered firm. Intuitively, the value of the unleveredfirm might happen to be lower than that of the levered firm if equityholders do not haveenough resources to achieve the first best level.17

17Note that Case 3 will arise however for only a very low liquidation value relative to a high value ofthe manager’s outside option.

The Valuation of Corporate Debt with Default Risk 21

4. DiscussionThe main objectives of our study were to construct a dynamic debt pricing model thatcomes close to the structure of most existing bankruptcy procedures; to explicitly incor-porate the inalienability of the human capital which is increasingly a key feature of themodern enterprise; and lastly but not least importantly to provide a clear distinctionbetween default and liquidation.The importance of the last objective cannot be overemphasized. It is clear that most

of the companies who default go into a period of reorganization and may or may not beliquidated. The time spent in the reorganization period varies immensely. Franks andTorous (1989) report that in their sample, firms on average spend a period of 4 years inChapter 11. Gilson, John and Lang (1990) find that in their sample only about 5% of thebankruptcies in Chapter 11 are converted into Chapter 7 liquidations. They report thatthis proportion varies from sample to sample and other studies have found that aboutone-third of the firms in Chapter 11 end up in liquidations. Franks and Sussman (2000)in their study on distressed UK companies find that banks use their control rights toencourage financially distressed firms to restructure. They find no evidence of automaticliquidation upon default. There are numerous examples of companies who pass througha period of reorganization after default but come out successful. It would therefore beincorrect to treat default and liquidation as being synonymous when modelling the leveredfirm.Our study does not only provide an arbitrage-free model for valuing equity and cor-

porate debt but also provides a framework to infer some interesting corporate financeimplications. In our framework both debtors and creditors can at any point in time bethe residual claimants. This is remarkably in contrast to the traditional “nexus of con-tracts” definition of the firm first provided by Jensen and Meckling (1976). As arguedby Rajan and Zingales (2000), in a world of incomplete contracts and multiple sources ofpower no party has predetermined payoffs and any party can at some point in time havea residual claim on the firm’s assets. This explains why maximisation of the value of theresidual claimant’s securities need not maximise the value of the firm as a whole.It is noteworthy that our model predicts that the “asset substitution” effects identified

by Jensen and Meckling (1976) and Green (1983) would be minimal.18 The asset substitu-tion problem arises when equityholders extract value from the debtholders by investing inrisky projects which might have a negative net present value. Such predatory behaviourwill not occur in our framework since the equityholders do not always have a residualclaim on the firm’s cash flows. Indeed in bad states of the world, the debtholders havemore bargaining power given the shift in the control rights on default. This explains whythe equityholders actually have a concave value function in the default region, while thedebtholders value function is convex in this region. (See Figure 2.) Indeed, Leland (1998)finds that the agency costs of debt related to asset substitution are insignificant.As argued by Jensen (1986) one motivation for issuing debt is to commit the managers

of the firm to pay out future cash flows. However for such an objective to be effectivethere has to be some punishment if the commitment is not fulfilled. This punishment cantake various forms. In our model it is the shift of the control rights that gives force tothis commitment. Managers realise that if a circumstance arises where they are unableto service the debt, then they will be left with a gross payoff that just equals the value oftheir outside option.18 I am grateful to Ronald Anderson for this observation.

The Valuation of Corporate Debt with Default Risk 22

Our framework also identifies the first best liquidation threshold which can then beused as a benchmark to compare the efficiency of the unlevered firm versus the leveredfirm. We find that the Modigliani-Miller theorem does not hold even in the absence of anycoordination problems, informational asymmetries, financial distress costs and taxes. Ourstudy finds that in general the value of the levered firm will be less than the value of theall equity firm as the creditors have an incentive to liquidate prematurely given an upperlimit to their payoffs. However depending on the parametric values, a situation might arisewhere the value of the levered firm is greater than the value of the all equity firm. Theintuition underlying this is that the all equity firm might be cash constrained because ofwhich the equityholders might have to liquidate earlier than they would have wanted to iftheir resource constraint had not been binding. Note that since the parametric values willdiffer from firm to firm and industry to industry, therefore, the efficiency of the liquidationthresholds vis-a-vis the first best liquidation point will also differ on a case by case basis.

5. Summary and ConclusionsWe have set up a debt pricing model with the following features: (a) the structure ofour model closely resembles the existing bankruptcy regimes; (b) the model incorporatesthe inalienability of human capital which is increasingly becoming a dominant feature ofthe modern firm; (c) a clear separation between default and liquidation is obtained; (d)the liquidation decision is made by the creditors following a default by the manager; (e)minimal asset substitution effects; (f) capital structure irrelevance not obtained even inthe absence of financial distress costs and taxes; (g) provides a framework to compare theefficiency of the liquidation thresholds of different firms. We have therefore provided aframework which is rich enough for valuing corporate securities and inferring some usefulimplications for corporate finance.

APPENDIX

Proof of Proposition 1: The technical problem one faces here is that of solving theordinary differential equation (6) subject to the boundary conditions (7), (8), (9) and

(10). If x <∧x, then e(x) = 0 and Eqn. (6) simply becomes the following second order

homogenous differential equation

1

2σ2x2E002 (x) + αxE02(x)− rE2(x) = 0 (29)

where E2 is the value of equity when x <∧x. It is easy to verify that the general solution

of Eqn. (29) is

E2(x) = N1xξ1 +N2x

ξ2 (30)

where N1 and N2 are the two integration constants and ξ1 and ξ2 are the positive andnegative roots respectively of the characteristic equation ξ (ξ − 1)σ2/2 + ξα − r = 0. If

x ≥ ∧x, then e(x) = x− a− b and the differential equation becomes

1

2σ2x2E001 (x) + αxE01(x)− rE1(x) + x− a− b = 0 (31)

where E1is the value of equity when x ≥ ∧x. The general solution to Eqn. (31) is

The Valuation of Corporate Debt with Default Risk 23

E1(x) = M1xξ1 +M2x

ξ2 +x

r − α −a

r− b

r(32)

where x/(r−α)−a/r− b/r is the particular solution andM1 and M2 are the integrationconstants. As x→∞, xξ1 explodes. Thus, given the no bubbles condition (7), M1 mustbe zero. We know that the value of equity is zero when x falls below x∗. We thereforeneed to determine the value of equity in the ranges x ∈ [x∗,

∧x) and x ∈ [

∧x,∞) which is

given by E2 and E1 respectively. We therefore now have three unknowns N1, N2 and M2

in Eqns. (30) and (32) and three equations given by the boundary conditions (8), (9) and(10). Solving for the three unknowns, yields the solution in Eqn. (11). Q.E.D.

Value of equity with riskless debt: If K ≥ b/r, then it will always be in the interest ofthe creditors to liquidate once a default occurs as they can then recover the total amountof their principal. The default point is therefore the same as the liquidation point in thiscase. Further the equityholders will be the residual claimants in all states of the world.Their payoffs are now given by

e(x) =

(x− a− b if x ≥ ∧

x

rK − b if x <∧x. (33)

It is obvious that when debt is riskless the value of equity when x <∧x is given by K−b/r.

We therefore need to determine the value of equity when x ≥ ∧x. The value of equity when

x ≥ ∧x will satisfy the following second order differential equation

1

2σ2x2E00(x) + αxE0(x)− rE(x) + x− a− b = 0.

The general solution of this equation is given by

E(x) = Q1xξ1 +Q2x

ξ2 +x

r − α −a

r− b

r

where Q1 and Q2 are the constants to be determined. The two boundary conditions thatwe need are

limx→∞E(x) =

x

r − α −a

r− b

r(34)

E³∧x´

= K − b

r(35)

The asymptotic no bubbles condition implies that Q1 = 0. Therefore we now have oneequation in one unknown. This readily yields the following solution for the value of equitywith riskless debt

E(x) =

(K − b

r if x <∧x

x−x1−ξ2

r−α − ar − b

r +¡K + a

r

¢x−ξ2 if x ≥ ∧

x. (36)

Proof of Proposition 2: Here we want to solve Eqn. (13) subject to boundary condi-tions (14), (15), (16), (17) and (18). When K < b/r, the value of debt is just equal to K

The Valuation of Corporate Debt with Default Risk 24

for x < x∗, as when the firm is liquidated the maximum that the debtholders can get isthe scrap value of the firm. Let the value of debt be C(x) for x ∈ [x∗,

∧x). Then given the

payoff function in Eqn. (12), Eqn. (13) becomes

1

2σ2x2C00(x) + αxC0(x)− rC(x) + θx− a = 0. (37)

It is easy to verify that the general solution to Eqn. (37) is of the following form

C(x) = A1xξ1 +A2x

ξ2 +θx

r − α −a

r(38)

where A1 and A2 are the two integration constants. Next consider the range x ∈ [∧x,∞).

Let B1 be the value of debt in this range. Now Eqn. (13) becomes

1

2σ2x2B001 (x) + αxB01(x)− rB1(x) + b = 0. (39)

The general solution to Eqn. (39) is

B1(x) = D1xξ1 +D2x

ξ2 +b

r(40)

where D1 and D2 are the integration constants. The asymptotic condition (14) impliesthat the coefficient D1 will be zero as ξ1 is positive. Therefore we now have four equations(15), (16), (17), (18) in four unknowns, A1, A2, D2 and x∗. Note that the free boundaryx∗ itself is an unknown. We solve for the value of debt in terms of x∗. Using Eqns. (15),(17) and (18) we solve for the unknown constants A1, A2 and D2 in terms of x∗. Thisyields the expression for B(x) given in Eqn. (19). Note that we are still left with oneunknown x∗ and one equation (16). However closer examination of Eqn. (16) reveals thatit is nonlinear in the threshold x∗. Thus x∗ can only be solved numerically from Eqn.(16). Finally, if K ≥ b/r, then debt is riskless and hence there is no default risk. Thusthe value of debt now just equals its principal value of b/r. Q.E.D.

Proof of Proposition 3: Again we need to solve an ordinary differential equation subjectto boundary conditions. The ODE is given in Eqn. (23) and the boundary conditions arestated in Eqns. (24), (25) and (26). We know that the value of the unlevered firm will beequal to K if x < L∗, where L∗ is defined in Eqn. (21). If x ≥ L∗, then given the payofffunction in Eqn. (22), the ODE becomes

1

2σ2x2V 00u (x) + αxV 0u(x)− rVu(x) + x− a = 0 (41)

the general solution of which is given by

Vu(x) = R1xξ1 +R2x

ξ2 +x

r − α −a

r(42)

where R1 and R2 are the constants to be determined. Given the asymptotic no bubblescondition we immediately know that R1 equals zero. We are now left with two equations(25), (26) in two unknowns R2 and x∗u. Recall that x∗u is a free boundary which deter-mines L∗ and which itself needs to be determined. Solving the two equations for the twounknowns yields the solution given in Eqns. (27) and (28). Q.E.D.

The Valuation of Corporate Debt with Default Risk 25

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